Multivariate density estimation under sup-norm loss: oracle approach, adaptation and independent structure

TL;DR

This paper introduces a data-driven multivariate density estimator under sup-norm loss that adapts to independence structures and solves bandwidth selection, with applications to anisotropic Nikolskii classes.

Contribution

It provides a fully data-driven estimator with oracle inequalities that accounts for independence structures and addresses bandwidth selection in multivariate density estimation.

Findings

Establishment of a sup-norm oracle inequality for the estimator.

The estimator adapts to independence structures in the data.

Application to adaptive estimation over anisotropic Nikolskii classes.

Abstract

The paper deals with the density estimation on Rd under sup- norm loss. We provide with fully data-driven estimation procedure and establish for it so called sup-norm oracle inequality. The pro- posed estimator allows to take into account not only approximation properties of the underlying density but eventual independence struc- ture as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in multivariate density model. Usefulness of the developed approach is illustrated by appli- cation to adaptive estimation over anisotropic Nikolskii classes.